A quick follow up for anyone who has posted a reply or is experiencing the same problem. I've found the root of the problem are after a lot of testing with different headphones.
The problem is caused by the type of earbud headphone - specifically earbud headphone that include a built-in mic. So this is not limited to the SGS4 earbuds, but rather earbuds that include a built-in mic.
Thus far I've tested with several different versions of Sony and Panasonic earbuds, and both Samsung S3 and S4 earbuds. So I'd expect it to be the same for all earbud headphones that include a built-in mic and are supplied along with the mobile phone.
Regular earbud headphones (without a mic) and any other stereo headphones work properly.
I have no idea why this happens with the note 3, but I hope this helps stop all the frustration felt by those who experiences this in future.